Use ID properties in cycles/nodes
You can store the face ID as a vertex color o each face then read the vertex color using the Attribute node.
This script will create a vertex color layer and store the face index as color :

import bpy

obj = bpy.context.object
obj_data = obj.data
group_name = 'face_ID'

#check for existing group with the same name
if None == obj_data.vertex_colors.get(group_name): 
    obj_data.vertex_colors.new(name=group_name)
color_map =  obj_data.vertex_colors[group_name]

#get faces' number 
count = len(obj_data.polygons)

for poly in obj_data.polygons:

    ID = poly.index / count  #for normalized results
    color = (ID, ID, ID)
    for loop_ind in poly.loop_indices:
        color_map.data[loop_ind].color = color